Apparatus and method to measure a molecular diffusion coefficient in a porous powder

ABSTRACT

An apparatus and method for measuring molecular diffusivity in porous powders or minerals, e.g., the microporous synthetic minerals used for gas separation and chemical catalysis, were invented that allows a varying pressure in the gas around the powder during the measurement. This obviates the need for the pressure to be constant and the attendant complicated component parts. A mass balance model equation of the sample cell with the diffusivity as an adjustable parameter is used to deduce the sample cell pressure change versus time (uptake curve) until the equilibration of gas diffusion into the powder. A numerical analysis method is used to solve the mathematical model to compute a simulated uptake curve. Curve fitting of the simulated and measured uptake curves is used to optimize the diffusivity parameter, which gives the measured diffusivity. The apparatus and method are simple, easy to use, and automation is also simple.

CROSS-REFERENCE TO RELATED APPLICATIONS

Priority is claimed pursuant to 35 USC 119a,b from People's Republic ofChina patent application number 201410515969.2, filed on Sep. 29, 2014.

BACKGROUND OF THE INVENTION

An aspect of this invention relates to the characterization of porouspowders of geological and synthetic minerals used in industrial gasseparation and solid catalyzed chemical processes wherein the speed ofmolecular diffusion in the porous powder is measured. In porous powders,different molecules diffuse at different speeds, that is, they havedifferent diffusion coefficients, and the difference in the diffusioncoefficients mainly determine how well they perform in the separation ofgases or as catalysts. The embodiments provide a simple method andapparatus for measuring a molecular diffusion coefficient in a porouspowder. The term “porous powder” is to be understood to refer to powdersthat comprise minerals that have porous cavities or channels and alsothose that comprise aggregates of nonporous minerals compressed to formporous particles. The term “diffusion coefficient” is also known as the“diffusivity”.

The speed at which molecules move into and out of a porous powder, whichis also known as the internal diffusion rate, can limit the reactionrate and control the selectivity when the powder is the catalyst of areaction. Knowledge of this diffusion rate can be used to guide themanufacture of a better catalyst. In another important application,differences in the diffusion rates can be utilized to separate thecomponents in a gas mixture. In this invention, the term “diffusion” isused in a narrower sense than in general to mean the internal diffusionof molecules inside a porous powder. Depending on the size of the poresin the powder, the molecules would diffuse as molecules in a gas phaseor when the pores are so small as to be of molecular dimensions andthere is no distinct gas phase, they diffuse as an adsorbed phase. Thediffusion coefficient for the molecular diffusion inside a porous powderis the parameter which is used to characterize the diffusion rate. Thus,the ability to easily carry out the quantitative measurement of thediffusion coefficient is highly advantageous for the development ofuseful porous powders.

The techniques for the measurement of a diffusion coefficient aredescribed in the monograph, Diffusion in Nanoporous Materials, by JorgKarger, Douglas M. Ruthven, and Doros N. Theodorou (Wiley-VCH, 2012),who divided these techniques into microscopic and macroscopictechniques. The microscopic techniques are based on the random walk orEinstein description of diffusion and they use the tracing of the pathof tracer molecules inside the pores to measure the diffusioncoefficient. However, because these techniques must trace particularmolecules, they must be capable of the detection of distinct moleculesand their molecular movement, and in general, they use expensiveinstrumentation and sophisticated procedures, e.g., the use of pulsedfield gradient nuclear magnetic resonance, quasi-elastic neutronscattering, interference microscopy or confocal fluorescence microscopy.These techniques are not suitable for an ordinary research laboratory orfor industrial use.

In a common research or industrial laboratory, the use of one of themacroscopic techniques is more reasonable. These have been based onsorption kinetics and a relaxation method which is usually a stepresponse method, where a step change from P₀ to P_(∞) is made in the gaspressure of the gas surrounding the porous powder and the curve of theconcentration of the molecules in the porous powder versus time (calledthe soption or uptake curve) is measured as the system relaxes to itsnew equilibrium state. In the prior art, most experimental techniquesmeasure the uptake curve by using a highly sensitive microbalance tomeasure the change in the weight of the powder versus time. However,this is disadvantageous because these microbalances are very expensive,and so there is an incentive to instead measure some changes in the gassurrounding the powder versus time as the response curve from which theuptake curve can be deduced, e.g., in the prior art, the gasreplenishment rate is used, but so far there has been no easy techniquefor this measurement. Such a response curve is also usually known as anuptake curve, so the method is also known as the uptake curve method.The diffusion coefficient is obtained by a curve fitting method with theuse of Fick's Second Law to describe the diffusion in the porous powder,namely, Eq. (1)

$\begin{matrix}{\frac{\partial q}{\partial t} = {D_{c}\frac{\partial^{2}q}{\partial x^{2}}}} & (1)\end{matrix}$

and its initial and boundary conditions

$\begin{matrix}{{t < 0},{P = P_{0}},{q = {q_{0}\left( {\forall x} \right)}}} & \left( {2a} \right) \\{{t \geq 0},{P = P_{\infty}},{{q_{x = R}} = q_{\infty}},{{\frac{\partial q}{\partial x}_{x = 0}} = 0}} & \left( {2b} \right) \\{{q_{0} = {f\left( P_{0} \right)}},{q_{\infty} = {f\left( P_{0} \right)}}} & \left( {2c} \right)\end{matrix}$

In Eqs. 1 and 2, P is the pressure, q is the molecular concentrationinside the porous powder particle, t is the time variable, x is thespace variable, D_(c) is the diffusion coefficient parameter, thesubscript 0 denotes initial condition, the subscript ∞ denotes the newequilibrium condition, R is the length of the diffusion pathway in theparticle, and q=f(P) denotes the functional form of the equilibriumrelationship between the concentration inside the particle and the gaspressure, which is also known as the adsorption isotherm. Briefly, Eqs.1 and 2 are used with an assumed value of the diffusion coefficientparameter to compute a simulated uptake curve which is compared with theexperimental uptake curve. The value of the diffusion coefficientparameter in Eq. (1) is then changed to optimize the fit between thesimulated and experimental uptake curves, with its value at the best fitbeing used as the measured diffusion coefficient.

In the prior art, the methods and apparatuses use an analytical solutionof Eqs. 1 and 2 to compute the simulated uptake curve. Since ananalytical solution is only available when the boundary condition whichcomprises the pressure of the gas environment surrounding the particleis a constant pressure, namely, Eq. 2b, therefore, the measurement hasto be made with a constant pressure of gas. That is, in these methodsand apparatuses, the gas environment of the particle has to be held at aconstant pressure. Thus, when a step response method is used, the changein the gas environment was a very rapid change from one constantpressure to another constant pressure. In most of the prior art, thesecond pressure is maintained constant by using a very large volume ofgas environment surrounding the powder so that the amount of gas thatdiffuse into the powder is a negligible fraction of it. However, thismeans that an auxiliary means must be used to measure the amount of gasthat has diffused into the powder, and most apparatuses use a highlysensitive microbalance, which has the disadvantage that such amicrobalance is very expensive and even then still does not have thevery high sensitivity desired. An example of this type of apparatus wasdescribed in a paper by Youngquist, Allen and Eisenberg (Industrial andEngineering Chemistry, Product Design Development, 10 (1971) 308).Another problem that the apparatus of this type faces is that its use ofa large static volume of gas that surround the powder sample gives riseto the question of whether the gas pressure is homogeneous. An apparatusto solve this latter problem made use of a new type of microbalance thatcan be used in a gas flow. This apparatus which used a flowing gassystem and a tapered element oscillating microbalance (TEOM) wasdescribed in a paper by Zhu, Kapteijn and Moulijn (Microporous andMesoporous Materials, 47 (2001) 157-171). However, the TEOM is a muchmore expensive microbalance and it is not suitable for use in a commonlaboratory or industrial laboratory.

Due to the disadvantage that available commercial microbalances are veryexpensive and yet still cannot give a really satisfactory sensitivityand resolution, there is an incentive to develop a technique to measuresome change in the gas phase surrounding the powder as the responsecurve instead of measuring the weight of the gas added into the powder.An example of this approach is the method and apparatus in U.S. Pat. No.4,762,010 to Borghard and Schoennagel, which used a flow controller thatwas capable of feeding in gas at very slow flow rates to replenish ormake up for the gas that has diffused into the powder in order to keepthe gas pressure constant. They used the measurement of the flow rate ofthe make-up gas to deduce the amount of gas that has diffused into thepowder versus time or the uptake curve. This patent teaches that it isnecessary to have gas feed rates that are very, very slow during themeasurement because the amount of gas diffusing into the powder is verysmall. It is evident that this need leads to many problems, such as (1)the need to have additional complicated component parts and proceduresto control the very, very slow replenishment flow rate needed tomaintain a constant pressure around the powder, (2) a very slow flowrate has to be measured, which was measured by a pressure change, butthe corresponding pressure change was extremely small, which made thecontrol of the flow rate very difficult and resulted in poor accuracy,and (3) in order to make the pressure change larger, the container thatsupplied the feed gas was made to be very small, but because it was verysmall, its supply of feed gas was limited and the measurement can onlybe made for a very small range of pressure. Thus, the apparatus has notbeen much used.

U.S. Pat. No. 6,981,426 to Wang, Wei and Wang teaches a method that usefewer additional complicated component parts, but this was achieved atthe expense of doing away with the automatic control of the requiredvery slow replenishing gas flow rate. However, because there was noautomatic control of the gas flow rate, the measurement procedure wasmade more complicated and tedious because a manual control of the gasflow rate had to be used, which is very inconvenient and laborintensive.

As discussed above, a basic difficulty that the prior art methods haveto face is the control of the required very slow replenishment ormake-up gas flow rate needed to maintain a constant pressure in the gassurrounding the powder sample. To avoid this difficulty, the more recentdevelopments in the measurement of internal diffusion coefficients haveturned to using apparatuses and methods that use a constant partialpressure of the measurement gas, instead of a constant total pressure,in the gas phase surrounding the powder sample so that the concentrationchange can be used to deduce the uptake curve. This is usually achievedwith the use of a flow system and the use of a constant ratio of flowrates of the gases. However, gas flow controllers can only be reliablyused for gas flow rates above 10 cm³ per minute. This together with thatit is desirable to carry out a diffusion measurement in the linearadsorption isotherm regime, which requires the partial pressure of themeasurement gas to be low and so its mole fraction in the gas phaseshould be only a few percent, requires that the total gas flow rateshould be a few hundred cm³ per minute. This is a high flow rate for alaboratory apparatus, and it leads to the problem of the production ofturbulence whenever there is a change in the flow rate of a componentgas, such that it is not possible to produce a sharp change in the gasmole fractions of the composition gases. This turbulence means that thegas phase composition is highly erratic and its measurement is quiteunreliable for some time after the step change had been made, whichincludes the period with much of the significant data for calculatingthe diffusion coefficient. One way of trying to solve this is to use thedetection of the weight change in the adsorbed phase, as was done in thework cited above by Zhu, Kapteijn and Moulijn who used a flow system anda TEOM. However, as pointed out above, the TEOM is a very expensivemicrobalance and not suitable for common use. Another way of solvingthis is to use only the data collected a long time after the step changehad been made, which was invented and reported in a paper by Eic andRuthven (M. Eic, D. M. Ruthven; Zeolites, 8 (1988) 40-45) and now knownas the long time limit zero length column (ZLC) method. However, thissolution is unsatisfactory because it has to assume that the powdersample is completely homogeneous, that is, the long time limit datagives the same diffusion coefficient as the short time limit data, whichis unlikely with most practical samples. The ZLC method has also beendeveloped to use the complete data set but the technique is difficultand it needs special instruction by the inventors to be able to collectthe data at short times after the step change, which is highlydisadvantageous for common use.

Yet another way of trying to solve the problem of the turbulence thatexist after a change in gas flow rate is to use some experimentalparameters to characterize its effects and then use these parameters tosubtract the effects of the turbulence, which was reported in a work byGuo and coworkers (Juhua Guo, Yuxin Li, Yanghuan Huang, Dezheng Wang;Journal of Nanoscience and Nanotechnology, 14(9) (2014) 6858-6864).Although their apparatus is simple and is basically that used in thechromatographic method, the preliminary work needed to characterize theturbulence is very time consuming, and hence disadvantageous. Thechromatographic method, on which this work by Guo and coworkers wasbased, is basically a long time limit method, which can only be usedwith highly homogeneous sample powders.

Other methods have also been devised to solve the shortcomings discussedabove, which include the frequency response method, which has beendescribed in a work by Yasuda and Yamamoto (Y. Yasuda, A. Yamamoto;Journal of Catalysis, 93(1) (1985) 176-181) and the temporal analysis ofproducts (TAP) method, which has been described in a work by Keipert andBaerns (O. P. Keipert, M. Baerns, Chemical Engineering Science, 53(20)(1998) 3623-3634). However, none of the methods has received much commonuse, and there is still a need for a simple and convenient method tomeasure the diffusion coefficient. The present invention meets thisneed. It is based on the discovery that in order to measure thediffusion coefficient in a porous powder, there is no need to make asharp change in the gas phase environment of the powder sample, and thata gradual change can be equally well used. This resulted in a means tomeasure the molecular diffusion coefficient in a porous powder that usedonly the simple apparatus and procedure used to make a gas adsorptionmeasurement. The measurement technique used in this invention is part ofthe art generally referred to as “physical modeling” in which thediffusion coefficient is measured by the optimization of the diffusioncoefficient parameter in the physical model where the criterion used inthe optimization is the best fit between the uptake curve calculated bythe physical model and the experimentally measured uptake curve.

BRIEF SUMMARY OF THE INVENTION

In accordance with one aspect of this invention, a method and apparatusfor measuring a molecular diffusion coefficient in a porous powdercomprise:

-   -   (a) providing a supply container and a sample cell of known        volume isolated by a gas flow control means, placing a weighed        amount of the porous powder in the sample cell, placing the        sample cell in a constant temperature bath, treating the porous        powder into a known chemical state, keeping the constant        temperature bath at a fixed temperature and evacuating by vacuum        pump means and then isolating the supply container and sample        cell;    -   (b) supplying a gas to the supply container, then using the gas        flow control means to allow communication between the supply        container and the sample cell for a period between one second to        thirty seconds, and measuring the pressure decrease and        temperature in the supply container and the pressure and        temperature in the sample cell as a function of time until        equilibration thereby determining a curve of sample cell        pressure versus time;    -   (c) using numerical analysis means to provide a computer program        further comprising a mass balance equation of the sample cell        with the diffusion coefficient as a parameter, and using a        computer with the computer program and an estimated value of the        diffusion coefficient and the measured pressure decrease,        temperature and volume of the supply container and measured        volume, temperature and weight of powder in the sample cell to        compute a curve of sample cell pressure versus time;    -   (d) repeating the computation in step (c) to optimize the        diffusion coefficient parameter by finding the best fit between        the computed curve of sample cell pressure versus time and the        corresponding curve measured in step (b), and using the        optimized diffusion coefficient as the measured diffusion        coefficient.

In accordance with one embodiment of this invention, an apparatus formeasuring a molecular diffusion coefficient in a porous powdercomprises:

-   -   (a) a supply container of known volume;    -   (b) a sample cell with a weighed amount of the porous powder and        of known volume and isolated from the supply container by a gas        flow control means, wherein the gas flow control means is in the        communicate state for between one second to thirty seconds;    -   (c) a constant temperature bath surrounding the sample cell;    -   (d) a vacuum pump assembly in communication with the supply        container and sample cell through valving means and connecting        conduits;    -   (e) at least one gas supply means in communication with the        supply container through valving means and connecting conduits;    -   (f) a pressure decrease measurement means in communication with        the supply container;    -   (g) a pressure measurement means in communication with the        sample cell;    -   (h) a temperature measurement means in contact with the supply        container;    -   (i) a temperature measurement means in contact with a tubing to        the sample cell;    -   (j) a computer and computer program, wherein the computer        program comprises a mass balance equation of the sample cell        with a diffusion coefficient as an adjustable parameter        formulated using numerical analysis means, and the computer        computes equilibration curves of the sample cell pressure versus        time using the computer program, a value of the diffusion        coefficient parameter and the measured pressure decrease,        temperature and volume of the supply container and measured        volume, temperature and weight of powder in the sample cell, and        the computer program further comprises an optimization means for        selecting the value of the diffusion coefficient parameter that        gives the best fit between the computed and measured curves of        the sample cell pressure versus time.

Accordingly, some advantages are that the measurement is simpler andmore accurate because, as distinct from prior art, there is no need tomaintain a constant pressure in the gas surrounding the powder duringthe measurements. The method and apparatus are simpler because thepressure in the gas surrounding the powder is allow to decreasenaturally, as would happen when some gas diffuse into the powder, ratherthan using some artificial means to force it to be roughly constant. Theexperimental results have a higher accuracy because of the use of areliably measured pressure change in the gas surrounding the powderinstead of a poorly controlled pressure that is roughly constant.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general schematic showing the modules of an apparatus usedto measure the molecular diffusion coefficient inside a porous powder.

FIG. 2 is a detailed schematic view of a first embodiment.

FIG. 3 is a detailed schematic view of a second embodiment.

FIG. 4 shows a graph of the data of the adsorption isotherm of Example1.

FIG. 5 shows graphs of the data used to calculate the moleculardiffusion coefficient of Example 1.

FIG. 6 shows an expanded view of the best-fit simulated and experimentalsample cell pressure versus time curves in FIG. 5.

In the drawings, the reference numerals are:

10—gas flow control means; 11—sample cell; 12—porous powder sample;13—supply container; 14—reference chamber; 16—gas supply; 17—gas supply;20—constant temperature bath; 21—pressure measuring means; 22—pressuremeasuring means; 23—pressure difference measuring means; 25—temperaturemeasuring means; 26—temperature measuring means; 27—temperaturemeasuring means; 31—vacuum pump assembly; 32—valve; 33—valve; 34—valve;35—valve; 36—valve; 37—valve; 40—data collection means; 41—computer;42—computer program; 201—experiment module; 202—measurement module;203—computation module.

DETAILED DESCRIPTION OF THE INVENTION First Embodiment

This embodiment measures the molecular diffusion coefficient inside aporous powder by supplying the molecules as a gas stream fed in for ashort time into a sample cell containing the powder. It is novel inproviding for that the gas stream is fed in for only a short time butthe feed is also not sudden and does not comprise a sharp opening andclosing of the gas flow control means. Although the opening and closingof the gas flow control means is for a quite short time, they should besteady to avoid any surge in pressure in the sample cell. After aninitial pressure increase due to the short time when there was a supplyof gas, the pressure of the gas surrounding the porous powder is thenallowed to decrease naturally due to gas diffusion into the porouspowder until equilibration is reached, that is, there is no furtherchange in the sample cell pressure. Its novel feature is in theprovisions made to obviate the need to maintain a constant pressure inthe sample cell. The principle of the measurement will be firstdescribed, then the apparatus, and finally the way to make ameasurement.

The principle of the measurement is based on the art referred to as“mathematical modeling” or “physical modeling”. A mathematical modelthat includes the diffusion process is constructed which contains thediffusion coefficient as a parameter, wherein the model is furtherdesigned so that when it is given a value for this parameter, it cancalculate a simulated uptake curve. The technique used to deduce thediffusion coefficient is popularly known as “curve fitting”, which isbasically an optimization procedure where the value of the diffusioncoefficient parameter in the physical model is adjusted to give the bestfit between the uptake curve calculated by the physical model and theexperimentally measured uptake curve. This is also the principle used inthe prior art in which the model used is Eqs. 1 and 2 presented above.An important aspect of the discussion above is that the use of the modelof Eqs. 1 and 2 brings with it the very difficult problem of the need tokeep a constant pressure or partial pressure of the measurement gas,which is because Eq. 2b comprises the condition that the porous powderhas a constant pressure or partial pressure of the measurement gas inits gas environment. The present invention is based on the discoverythat with an auxiliary equation of a mass balance in the sample cell,the mathematical model of the diffusion process in the porous powderdoes not need to have a constant pressure or partial pressure of themeasurement gas in its gas environment. Thus, in the measurement of thediffusion coefficient in a porous powder, there is then no need to makea sharp change in the pressure of the gas around the powder sample, anda gradual change in the pressure can be used. This results in a means tomeasure the molecular diffusion coefficient in a porous powder thatneeds to use only the simple apparatus and procedure for a gasadsorption measurement with the addition of a mathematical model andcurve fitting procedure.

The apparatus and method to measures the molecular diffusion coefficientinside a porous powder is now described. The apparatus comprises thethree modules shown in FIG. 1, namely, an experiment module 201, ameasurement module 202 and a computational module 203. The components ofthe experiment module and measurement module are shown in FIG. 2. Theexperiment module comprises supply container 13, sample cell 11 andporous powder sample 12, gas flow control means 10, constant temperaturebath 20, vacuum pump assembly 31, and gas supplies 16 and 17. Themeasurement module comprises pressure measuring means 21 and 22,temperature measuring means 26 and 27, and data collection means 40. Thecomputational module comprises a computer 41 and computer program 42.The experiment module is used to prepare the system for a measurement,and comprises first evacuating and then filling supply container 13 withthe measurement gas, evacuating sample cell 11, and then opening gasflow control means 10 for a brief period to allow a transient flow ofgas from the supply container into the sample cell to initiate ameasurement. The measurement module is used to measure the pressuredecrease and temperature in the supply container until the gas flow isstopped and the pressure and temperature in the sample cell as afunction of time until equilibration is reached. The computationalmodule is used to apply an optimization technique to get the diffusioncoefficient by seeking the best fit between the calculated and measuredsample cell pressure versus time (uptake) curves. Data collection means40 is an analog to digital (A/D) conversion board placed in computer 41.It receives data signals from the sensors of pressure measuring means 21and 22 and temperature measuring means 26 and 27, which are convertedinto digital data that are saved in computer 41. The data collectionmeans can also be a person who reads the meter readings of the sensorsand then records these readings into the computer. This embodiment usesMKS Baratron Model 120AA capacitance manometers as the pressuremeasuring means in order to have a higher measurement precision. Thismodel has a measurement precision which is less than 0.3% of the readingshown by the measuring means. Other types of manometers or multiplemanometers with different ranges can also be used. After datacollection, computational means 203 is used to deduce the measureddiffusion coefficient by a curve fitting techniques using the calculatedand measured uptake curves.

The measurement proceeds as follows. A weighed amount of about 0.1 g ofporous powder sample 12 is placed in sample cell 11. The sample has tobe first treated to get it into a known chemical state, e.g., a degassedand dehydrated state for a zeolite, before the measurement, which iscalled a pretreatment, but before performing this pretreatment, thevolume of the sample cell is measured using the expanded pressure thatresults from the supply of a known amount of helium gas. This volumemeasurement is described in textbooks on gas adsorption and so this willnot be discussed here. The volume of the sample cell can alternativelybe measured after the diffusion coefficient measurement. Sample cell 11is connected to gas flow control means 10 by a tubing and the volume ofthe sample cell should be understood to include the volume of all thetubings, e.g., in FIG. 3, this volume would include the volumes insidethe pieces of tubing to gas flow control means 10, valve 32 and pressuremeasuring means 21 and the volumes inside these components. In manymeasurements, powder sample 12 is placed in a constant temperature bathat a different temperature from room temperature, and in these cases,the gas volume should include the effect of the temperature and if thegas is non-ideal at this temperature, a correction should be made forthis. For some samples where the pretreatment requires a flowing gas,the sample cell should be modified to allow a gas flow (not shown inFIG. 2). The pretreatment procedure depends on the nature of the sampleand can differ widely, but would be well known to someone skilled in theart and so this will not be discussed here. For example, with a catalystsample, the aim of the pretreatment would be to get the sample into acatalytically active state. For the pretreatment, the part of the samplecell that contains the powder sample is placed in constant temperaturebath 20, which can be a liquid constant temperature bath or anelectrical furnace controlled to be at a constant temperature. After thepretreatment, with gas flow control means 10 shut, vacuum pump assembly31 is used to evacuate the sample cell and sample through valve 32 andconstant temperature bath 20 is set to the measurement temperature. Atthe same time, the vacuum pump assembly is also used to evacuate supplycontainer 13 through valve 33. The vacuum pump assembly is aturbo-molecular pump and mechanical pump combination, but other pumpcombinations for gas evacuation can also be used.

The other arm of gas flow control means 10 is connected by tubing tosupply container 13, which is being prepared to be used to supply themeasurement gas. The volume of the supply container would have beenmeasured and so it is a known quantity; otherwise it has to be measuredafter the diffusion coefficient measurement. This volume should beunderstood to include the volume of the tubings, e.g., in FIG. 2, thisvolume would include the volumes inside the pieces of tubing to gas flowcontrol means 10, valves 33, 34, and 35, and pressure measuring means 22and the volumes inside these components. Pump assembly 31 is used toevacuate supply container 13 through valve 31. If a previous measurementhad used the same gas and the gas in the supply container is still pure,the evacuation can be omitted. After evacuation, gas supply 16 and valve35 or gas supply 17 and valve 34 is used to fill supply container 13with the measurement gas to a chosen pressure, which is chosen byexperience to have enough gas to complete a series of measurements. Thegas supply can be a high pressure gas cylinder with a low-pressurepressure controller and gauge or a glass tube filled with a liquid keptat a constant temperature for gases that are liquid at room temperatureand atmospheric pressure. More gas supplies can be added to theapparatus if it makes the apparatus more convenient to use.

After the supply container has been filled with the gas and the sampleis at the measurement temperature and the sample cell has beenevacuated, valve 32 is shut to isolate the sample cell. Then gas flowcontrol means 10 is opened briefly to supply a controlled amount of thegas to the sample cell. Gas flow control means 10 is a fine control leakvalve which is manually controlled. This is the type of leak valve thatis used to dose very small amounts of a gas into a vacuum system and canbe bought from suppliers of vacuum parts. Other types of flow controlvalves, such as a manual needle valve or a computer controlled solenoidvalve or servomotor valve, can also be used. The gas flow control meansis used to control the amount of gas supplied to the sample cell so thatthere is only a small pressure increase. The amount of gas supplied isdetermined by the nature of the experiment and experience of theoperator. In our measurements, the amount is chosen to allow 10 to 20doses of the gas to be made during the overall pressure increase in thesample cell from vacuum to the adsorption saturation pressure of thesupplied gas. These would give data collected at 10 to 20 differentpressure ranges, from which the values of the diffusion coefficient canbe obtained for different pressure or adsorbate concentration. Althoughgas flow control means 10 is to be opened or kept in its communicatestate for only a brief period, which is from 1 to 30 seconds, the valveis to be open in a steady manner, which is not to be a very sudden openand shut process that takes less than one second.

After gas flow control means 10 is opened, the pressure in supplycontainer 13 decreases. The pressure decrease is measured by subtractingthe pressure measured by pressure measuring means 22 at any time t fromthe pressure measured before gas flow control means 10 was opened. Thispressure decrease is recorded by data collection means 40 as a functionof time until after gas flow control means 10 is shut. After gas flowcontrol means 10 is opened, the pressure in sample cell 11 firstincreases, and then after the gas flow control means is shut, thepressure decreases due to diffusion into powder sample 12. The pressureis measured by pressure measuring means 21 and recorded by datacollection means 40 as a function of time until the pressure does notchange, that is, until equilibration was reached. The recorded data ofsample cell pressure versus time comprise the uptake curve, which isused for curve fitting to deduce the diffusion coefficient. Due to thatthe pressure is affected by the temperature, temperature measuring means26 and 27 are used to measure, respectively, the temperatures of supplycontainer 13 and the tubing leading to sample cell 11 so that the effectof any temperature change can be accounted for. This completes one cycleof data collection which can be used to determine the diffusioncoefficient in this range of adsorbate concentration. The adsorbateconcentration is the amount of molecules in the porous powder per gramof powder. After the sample cell pressure does not change any more, themeasurement can be repeated to determine the diffusion coefficient inanother range of adsorbate concentration, which can be used to see if itchanges with the adsorbate concentration. If the pressure in the supplychamber is sufficiently higher than the pressure in the sample cell,there is no need to refill the supply chamber for this. The measurementscan be repeated until the pressure in the sample cell has reached apressure beyond which no additional gas uptake occurs in the porouspowder, which is its saturation pressure. In addition to the diffusioncoefficient, those skilled in the art would also see that these datacomprise the data of adsorbed amount versus pressure, which is theadsorption isotherm. These data can be curve fitted to determine theparameters in the adsorption isotherm, which is an equation thatexpresses the thermodynamic equilibrium relationship between the gaspressure and adsorbate concentration. This adsorption isotherm can thenbe used in Eqs. 14b and 14c (see below) of the mathematical model usedto determine the diffusion coefficient.

Computational module 203 is used with an initial estimate of thediffusion coefficient parameter in Eq. 13 (see below) together with themeasured pressure decrease and temperature and known volume of thesupply container and measured volume, temperature and mass of powder inthe sample cell to calculate a curve of sample cell pressure versus time(uptake curve). Then the value of the diffusion coefficient parameter ischanged in order to optimize the fit between the computed uptake curveand the measured uptake curve. The optimized diffusion coefficientparameter that gives the best fit is taken to be the measured diffusioncoefficient.

From the description above, it can be seen that after the opening of gasflow control means 10, there is no adding of make-up gas into samplecell 11. This is different from the prior art method, that is, themethod of this invention avoids the difficult supplying of a very slowcontinuous flow of make-up gas and instead it just needs the gas flowcontrol means to be open for a brief period, which makes the method andapparatus simpler and more practical. This is because the mathematicalmodel is formulated to allow the use of a variable pressure in theboundary condition that expresses the adsorption equilibrium between thegas pressure and adsorbate concentration at the surface of the porouspowder, which is the adsorption isotherm (see Eqs. 14b and 14c below).In addition, in the technique in the prior art, due to the need tosupply a continuous flow of make-up gas and because this gas flow isvery, very small, the volume of the supply container has to be alsosmall in order to have a measureable change in its pressure. However,when this chamber is small, although the pressure change will be largeand can be reasonably measured, the pressure quickly falls to be closeto that of the pressure in the sample cell and so there is a need toreplenish the gas in the supply container many times. Due to that anerror is introduced each time a replenishment occurs, the overallaccuracy will be adversely affected. In contrast, in the presentinvention, the supply container can be quite large and it containsenough gas to complete the whole series of measurements, and therefore,the accuracy is better.

Detailed Description of a Second Embodiment

FIG. 3 shows the schematic of a second embodiment of the apparatus. Thedifference between this and the first embodiment is that the pressuredecrease in supply container 13 is measured with pressure differencemeasuring means 23, which has its low pressure part connected to thesupply container and the high pressure part connected to referencechamber 14. Reference chamber 14 is filled with a gas to the same or ahigher pressure than the pressure in the supply container and then keptisolated so that its pressure does not change. When gas flow controlmeans 10 is opened briefly, some gas flows out from supply container andits pressure decreases. Since there is no change in the pressure inreference chamber 14, the pressure difference recorded with pressuredifference measuring means 23 at any time t minus the pressuredifference measured before gas flow control means 10 was opened is thepressure decrease in the supply container at time t. This method ofmeasuring the pressure decrease will give a much higher measurementprecision when pressure difference measuring means 23 has a precisionwhich is proportional to its reading, e.g., a MKS Baratron Model 120ADdifferential pressure capacitance manometer, which has a measurementprecision less than 0.3% of the reading shown by the measuring means.Other types of differential pressure manometers or multiple differentialpressure manometers with different ranges can also be used.

Most of the operations performed with embodiment 2 are the same as withthe first embodiment, and only operations that are different will bedescribed. These involve how to make use of reference chamber 14. Vacuumpump assembly 31 is first used to evacuate the reference chamber at thesame time as when supply container 13 is evacuated by opening valves 33,36 and 37. Then valve 33 is shut to isolate them from the vacuum pump.With gas flow control means 10 closed, gas supply 16 and valve 35 or gassupply 17 and valve 34 is used to supply reference chamber 14 and supplycontainer 13 with the measurement gas to a chosen pressure, which ischosen by experience to have enough gas to complete a series ofmeasurements. Reference chamber 14 is then isolated by shutting valve37, and supply container 13 is used to supply gas to the sample cell asin embodiment 1. The pressure difference reading from pressuredifference measuring means 23 is sent to data collection means 40 andrecorded in computer 41. Due to that the pressure is affected by thetemperature, temperature measuring means 25 is used to measure thetemperature of reference chamber 14 so that the effect of anytemperature change can be accounted for. In FIG. 3, it is also shownthat reference chamber 14 and supply container 13 are connected by aslab of stainless steel which allows heat transfer but not mass transferbetween them, which is used to minimize any temperature differencebetween reference chamber 14 and supply container 13.

Example Measurement Using Embodiment 2

The measurement of the diffusion coefficient is now described with anexample. This example measured the diffusion coefficient of propaneinside a SAPO-34 zeolite sample. First, the free volume in sample cell11, which is the volume in the sample cell not occupied by the sample,is determined. It is presumed that the volume of the supply container isalready known. During a series of measurement of the diffusioncoefficient with different sample cell pressures, the adsorptionisotherm, which is the equation expressing the equilibrium relationshipdependence of the adsorbate concentration on the pressure, can also bemeasured. If the adsorption isotherm is already available from a paperin the literature, this can be used instead. These measurements aredescribed in textbooks on gas adsorption and so their details will notbe described here. FIG. 4 shows the adsorbate or adsorbed phaseconcentration versus pressure data and the best-fit curve of theadsorption isotherm, which was the equation

$\begin{matrix}{q = {{q_{sat}^{1}\frac{b_{1}P}{1 + {b_{1}P}}} + {K_{1}P}}} & (3)\end{matrix}$

where q, q_(sat) ¹, b₁, K₁, and P are the adsorbate concentration, acidsite concentration of the sample, adsorption equilibrium constant on theacid sites, adsorption equilibrium constant on the nonacid sites and gasphase pressure in the sample cell, respectively. The fitted values ofthe parameters were: q_(sat) ¹=0.70 mmol/g, b₁=0.36 l/Pa, and K₁=0.0051mmol/g/Pa. q and P have units of mmol/g and Pa. Eq. 3 is one example ofan adsorption isotherm, which is an equation expressing thethermodynamic equilibrium adsorbate concentration on the outer surfaceof a powder particle as a function of the sample cell pressure. Othergases adsorbed on other porous powder can have different adsorptionisotherm equations to describe the equilibrium relationship between thegas and adsorbed phases. Eq. 3 is used in Eqs. 14b and 14c (below),which comprise the boundary condition for Eq. 10. Alternatively, if anadsorption isotherm equation is already known from the literature, thiscan be used or if the adsorption data are too sparse to allow a curvefitting procedure, the parameters in the adsorption isotherm equationwould need to be treated as additional parameters to be fitted forduring the curve fitting to get the diffusion coefficient.

Many cycles of measurement can be made where in each cycle a curve ofthe sample cell pressure versus time (uptake curve) is used for curvefitting to determine the diffusion coefficient in a particular range ofpressure. In this example, FIG. 5 shows the data for when the samplecell pressure is about 30 torr. The following symbols and values areused:

V1 is the volume of supply container 13, V1=648.5 cc (previouslymeasured);

T1 is the temperature of supply container 13, T1=29.0° C. (did notchange with time during the measurement);

T2 is the temperature of the part of sample cell 11 that is not in theconstant temperature bath, T2=29.0° C. (did not change with time duringthe measurement);

Vu is the volume of the part of sample cell 11 at room temperature,Vu=10.35 cc;

Vc is the volume of the part of sample cell 11 in the constanttemperature bath, Vc=62.2 cc;

T3 is the temperature of the constant temperature bath, T3=29.0° C. (didnot change with time during the measurement);

dP is the pressure difference between reference chamber 14 and supplycontainer 13 (changes with time during the measurement);

P is the pressure in sample cell 11 (changes with time during themeasurement);

N is the number of moles of gas in the gas phase in sample cell 11(changes with time during the measurement);

m_(cat) is the weight of the porous powder sample in sample cell 11,m_(cat)=0.10 g;

F is the molar flow rate of the gas into sample cell 11 (changes withtime during the measurement).

FIG. 5 shows an example of the data collected when gas flow controlmeans 10 was open for a brief period of about 13 seconds to give atransient flow of gas into sample cell 11. It can be seen from the uppercurve that there is first an increase in the sample cell pressure due tothe gas fed into it, and then after the gas feed was shut off, there isa gradual decrease as gas diffuse into the porous powder sample. Duringthe measurement, the sample cell pressure changed, which distinguishesthis invention from the methods in the prior art. The significance ofthis is that it is advantageous because this is how the natural processoccurs, and there is no need to add complicated component parts andprocedures to prevent the natural pressure decrease as the gas diffuseinto the powder. In addition, in order to have the best accuracypossible, provisions were made to measure the pressure in sample cell 11and pressure decrease in supply container 13 with high precision meansthat have measurement precision that were less than 0.3% of the reading.A further consideration is that the volumes of sample cell 11 and supplycontainer 13 have to be chosen well. For 0.1 g of a porous powder samplewith a specific surface area of 300 m²/g, the volumes of sample cell 11and supply container 13 can be chosen to be about 70 and 650 cm³,respectively. FIG. 5 shows that after the brief flow of gas from thesupply container to the sample cell, the pressure decrease in the supplycontainer was 2.5 torr. That is, pressure difference measuring means 23showed a pressure difference of 2.5 torr. Since the precision is 0.3% ofthe reading, the measurement error was 0.003×2.5=±0.0075 torr, whichshowed that the pressure difference was quite precisely measured.Furthermore, 20 doses of gas where each dose gave about 2.5 torrpressure decrease would give a total pressure drop of about 50 torr. Ifthe supply container was originally filled to a pressure of 900 torr, itwould still have 850 torr left, which is sufficient to supply gas to thesample cell until atmospheric pressure. Usually, it is convenient tohave a supply container with a gas amount sufficient to increase thepressure in the sample cell to a pressure beyond which no additional gasuptake occurs in the porous powder. From FIG. 5, it is seen that thesample cell pressure decreased by about 4 torr after gas flow controlmeans 40 was shut, which was due to diffusion into the sample. Thepressure in the sample cell during this process was about 30 torr, andsince the precision is 0.3% of this, the measurement error was ±0.09torr, thus the pressure decrease of 4 torr could be measured quiteprecisely. However, if the pressure in the sample cell during thisprocess had been about 500 torr, the measurement error would be ±1.5torr, which would be quite large compared to a pressure decrease of 4torr. If this is a pressure range that is of interest, a smaller volumesample cell should be used so that the pressure decrease due todiffusion into the sample can be larger. In summary, the volumes ofsample cell 11 and supply container 13 should be chosen according to themeasurement needs and operator experience. From the data above,recommended volumes are between 200 and 500 cc for the supply containerand between 1 and 100 cc for the sample cell, with the smaller volumegiving better accuracy and the larger volume giving more convenience.Similarly, the brief period when the gas flow control means is open canbe recommended to be between 1 and 30 second, with the shorter timegiving more data points and the longer time giving more easilymeasurable pressure changes.

The procedure to get the diffusion coefficient is as follows. Amathematical model of mass balance in the sample cell is set up thatcontains the diffusion coefficient as an adjustable parameter. Numericalanalysis is used to write the equations of the model into a form that issuitable for use in a computer program, and which a computer can run tocalculate a simulated uptake curve. The diffusion coefficient parameteris optimized to get the best fit between the simulated and measureduptake curves. The mass balance equation for the sample cell is

$\begin{matrix}{\frac{N}{t} = {{{- x} \cdot m_{cat}} + F}} & (4)\end{matrix}$

In Eq. 4, t is the time variable, x is the molar flux per gram sample ofgas diffusing into the powder sample, which is a function of time, thatis, it is x(t), and F is the molar flow of gas into the sample cell,which is also a function of time, that is, it is F(t). x(t) iscalculated from Eq. 15 (below). This model, that is, Eq. 4, is anequation that describes the mass balance in the sample cell as that theaccumulation of moles of gas is equal to the loss due to diffusion intothe powder and the gain due to flow into the sample cell.

In order to have a wide use, from the consideration that under manyconditions, x(t) does not have an analytic formula but is onlycalculable in tabular form as values of x at different values of t, itis preferable to solve Eq. 4 by numerical analysis. The term “numericalanalysis” should be understood to mean that Eq. 4 is expressed as adiscretized formula in place of the differential equation, which can beby finite difference means, finite element means, finite volume means orspectral method means. One particular way to discretize Eq. 4 is by theEuler method, which is a first order finite difference method, whichgives

$\begin{matrix}{\frac{N_{i} - N_{i,1}}{\Delta \; t} = {{{- x_{i - 1}} \cdot m_{cat}} + F_{i - 1}}} & (5)\end{matrix}$

In Eq. 5, the subscript i denotes the time index of the discretized timevariable t. At is the time step size, which has to be small enough forEq. 5 to be an accurate approximation of Eq. 4. In the calculations inthis example, Δt=0.1 s. Eq. 5 is solved by a time march method, namely,

N _(i) =N _(i-1) −x _(i-1) ·m _(cat) ·Δt+F _(i-1) ·Δt  (6)

The Euler method illustrated here is an example and some other numericalanalysis method can also be used.

The third term on the right hand side of Eq. 6 is calculated from thedata of the pressure decrease in supply container with the use of theideal gas law by

$\begin{matrix}{{{F_{i - 1} \cdot \Delta}\; t} = {{\Delta \; F_{i - 1}} = \frac{\left( {{dP}_{i} - {dP}_{i - 1}} \right)V_{1}}{{RT}_{1}}}} & (7)\end{matrix}$

In Eq. 7, dP is a function of t, that is, it is dP(t), and dP_(i) is thediscretized form. The times of the measured values of dP recorded in thecomputer did not match exactly with the discretized time steps. In thiscase, an interpolation technique was used to get the values at therequired discretized times using the measured dP(t)˜t curve. The valuesof x(t) or x_(i) was obtained from the solution of the diffusionequation in the porous powder sample. The general form of this equation,which is also known as Ficks second law, is given in the monograph byCrank (J. Crank, The Mathematics of Diffusion, second edition, OxfordUniversity Press, 1975) as

$\begin{matrix}{\frac{\partial q}{\partial t} = {{div}\left( {D\mspace{14mu} {grad}\mspace{14mu} q} \right)}} & (8)\end{matrix}$

which in terms of orthogonal axes is

$\begin{matrix}{\frac{\partial q}{\partial t} = {{\frac{1}{A_{\xi}}\frac{\partial\;}{\partial\xi}\left( {A_{\xi}D_{\xi}\frac{\partial q}{\partial\xi}} \right)} + {\frac{1}{A_{\eta}}\frac{\partial\;}{\partial\eta}\left( {A_{\eta}D_{\eta}\frac{\partial q}{\partial\zeta}} \right)} + {\frac{1}{A_{\zeta}}\frac{\partial\;}{\partial\zeta}\left( {A_{\zeta}D_{\zeta}\frac{\partial q}{\partial\zeta}} \right)}}} & (9)\end{matrix}$

In Eq. 9, q is the adsorbate concentration or molecular concentrationinside the powder particle and it is a function of t and the spacevariables. A and D are the cross-sectional area and diffusioncoefficient, respectively, in the direction of the subscript, ξ, η, orζ, which denote the three generalized orthogonal space coordinates. Whenthis equation is solved by numerical analysis, A and D can be functionsof the space variables. This equation can be solved by the alternatingdirection implicit (ADI) method. Since the principle is the same forparticles of different geometries or shapes, the method of solution toget q is illustrated below with a spherical particle.

For a spherical particle, due to the symmetry of a sphere, only theradial coordinate will have a concentration variation and Eq. 9 issimplified to

$\begin{matrix}{\frac{\partial q}{\partial t} = {\frac{1}{r^{2}}\frac{\partial\;}{\partial r}\left( {r^{2}D_{r}\frac{\partial q}{\partial r}} \right)}} & (10)\end{matrix}$

In Eq. 10, r is the radial coordinate. Using the Crank-Nicolson methodand j−1, j, j+1 and n, n+1 to represent discretized r and t, Eq. 10 indiscretized form is

$\begin{matrix}{{\frac{q_{j}^{n + 1} - q_{j}^{n}}{\Delta \; t} = {{0.5\frac{1}{r_{j}^{2}}\frac{{r_{j + \frac{1}{2}}^{2}{D_{j + \frac{1}{2}}\left( {q_{j + 1}^{n + 1} - q_{j}^{n + 1}} \right)}} - {r_{j - \frac{1}{2}}^{2}{D_{j - \frac{1}{2}}\left( {q_{j}^{n + 1} - q_{j - 1}^{n + 1}} \right)}}}{\left( {\Delta \; r} \right)^{2}}} + {0.5\frac{1}{r_{j}^{2}}\frac{{r_{j + \frac{1}{2}}^{2}{D_{j + \frac{1}{2}}\left( {q_{j + 1}^{n} - q_{j}^{n}} \right)}} - {r_{j - \frac{1}{2}}^{2}{D_{j - \frac{1}{2}}\left( {q_{j}^{n} - q_{j - 1}^{n}} \right)}}}{\left( {\Delta \; r} \right)^{2}}}}}\mspace{79mu} {where}\mspace{79mu} {{D_{j + \frac{1}{2}} = {\frac{1}{2}\left( {D_{j + 1} + D_{j}} \right)}},\mspace{79mu} {r_{j + \frac{1}{2}} = {\frac{1}{2}\left( {r_{j + 1} + r_{j}} \right)}}}} & (11)\end{matrix}$

with a corresponding meaning for other subscripts with a ½ value. Eq. 10is not defined at r=0 because of the 1/r² term, and l'Hospital's rulehas to be used there, which gives

$\begin{matrix}{{\lim\limits_{r\rightarrow\infty}\left( {u_{rr} + {\frac{2}{r}u_{r}}} \right)} = {{u_{rr} + {\lim\limits_{r\rightarrow 0}{\frac{2}{r}u_{r}}}} = {{u_{rr} + {2u_{rr}}} = {3u_{rr}}}}} & (12)\end{matrix}$

This and the symmetry boundary condition at the center of the sphere,

${{\frac{\partial q}{\partial r}_{r = 0}} = 0},$

were used to write the discretized equation at r=0 as

$\begin{matrix}{\frac{q_{0}^{n + 1} - q_{0}^{n}}{\Delta \; t} = {{3D_{0}\frac{\left( {q_{1}^{n + 1} - {2q_{0}^{n + 1}} + q_{- 1}^{n + 1}} \right)}{(\Delta)^{2}}} = {3D_{0}\frac{2\left( {q_{1}^{n + 1} - q_{0}^{n + 1}} \right)}{\left( {\Delta \; r} \right)^{2}}}}} & (13)\end{matrix}$

Here, D₀ denotes the diffusion coefficient at r=0 (center of thesphere). Using P to denote the gas phase pressure in the sample cell,and P₀ and P(t) as the pressure before and after gas flow control meanswas opened, the initial and boundary conditions for Eq. 10 are

$\begin{matrix}{{t < 0},{P = P_{0}},{q = {q_{0}\left( {\forall r} \right)}}} & \left( {14a} \right) \\{{t \geq 0},{P = {P(t)}},{{{q(t)}_{r = R}} = q_{R}^{n}},{{\frac{\partial q}{\partial r}_{r = 0}} = 0}} & \left( {14b} \right) \\{{q_{0} = {f\left( P_{0} \right)}},{q_{R}^{n} = {f(P)}}} & \left( {14c} \right)\end{matrix}$

In Eq. 14, R is the radius, and the subscript 0 denotes the initialconditions at t<0. At t≧0, as shown in Eq. 14b, the pressure P is afunction of t, that is, P=P(t). This is the simulated uptake curve, thatis, the change in the calculated sample cell pressure with time. In Eq.14c, the equations q₀=f(P₀), q_(R) ^(n)=f(P) denote the equilibriumrelationship between the adsorbate concentration on the outer surface ofthe particle and pressure in the gas phase, which is the adsorptionisotherm (equation). This was Eq. 3 in this example, but it can be someother equation in other situations. Taken together, Eqs. 14b and 14c saythat at any time t (which is indicated by the discretization index n)when the pressure in the sample cell is P, the adsorption isotherm isused to calculate the adsorbate concentration on the outer surface ofthe particle, q_(R) ^(n). The use of the mass balance equation for thesample cell, which is Eq. 6 in this example, means that the pressure inthe sample cell, P, does not have to be constant because Eq. 6 gives itsvalue as a function of time.

Eqs. 12, 13, and 14 were solved by the Crank-Nicolson method, which isexplained in the monograph by Crank cited above. This method is based onthat Eqs. 12 and 13 form a tri-diagonal matrix, which can be solved withthe Thomas algorithm. In this example, the step sizes used were Δt=0.1 sand Δx=1×10⁻⁶ cm (the radius of the spherical particle was 1×10⁴ cm).This method used is an example and other numerical analysis methods andstep sizes can also be used. After each time step, the concentrationinside the particle q is known as a function of the radial coordinate,thus the concentration gradient at the particle surface can becalculated from which x(t) is calculated using

$\begin{matrix}{{x(t)} = {{{\frac{4\pi \; R^{2}}{\frac{4}{3}\pi \; R^{3}}\rho \; D_{R}\frac{\partial q}{\partial r}}_{r = R}} = {{\frac{3\rho}{R}D_{R}\frac{\partial q}{\partial r}}_{r = R}}}} & (15)\end{matrix}$

In Eq. 15, ρ is the density of the porous powder sample. The use of x(t)as x_(i-1) in Eq. 6 allows the value of N_(i) to be calculated, which isconverted into the sample cell pressure using Eq. 16:

$\begin{matrix}{N = {\frac{{PV}_{u}}{{RT}_{2}} + \frac{{PV}_{c}}{{RT}_{3}}}} & (16)\end{matrix}$

In Eq. 6, N_(i) is the discretized form of N as a function of t, thatis, it is N(t) and from Eq. 16, P is obtained also a function of t, thatis, it is P(t), which is the simulated uptake curve that is used for thecurve fitting with the measured uptake curve.

Eq. 10 is the diffusion equation inside a spherical particle. Otherparticle geometries, e.g., slab, cube, rectangular cube, etc., can besimilarly solved to give simulated uptake curves. FIG. 6 shows for thisexample the simulated uptake curve after the optimization of thediffusion coefficient parameter to give the best fit with the measureduptake curve. The best fit curve was obtained with a cube geometry forthe particle and the value of the diffusion coefficient parameter thatgave this simulated uptake was 0.70×10⁻¹⁶ m²s⁻¹. In this example, thecurve fitting was performed visually, that is, the curves were comparedby eye and the operator decides which curve gave the best fit. Othermethods to optimize the diffusion coefficient can also be used.

What is claimed is:
 1. A method and apparatus for measuring a moleculardiffusion coefficient in a porous powder comprising: (a) providing asample cell that has a weighed amount of the porous powder and a supplycontainer of known volumes isolated by a gas flow control means, placingthe sample cell in a constant temperature bath and treating the porouspowder into a known chemical state, keeping the sample cell in theconstant temperature bath at a fixed temperature and evacuating thesample cell and supply container by vacuum pump means and then isolatingthe supply container and sample cell; (b) supplying a gas to the supplycontainer, then using the gas flow control means to allow communicationbetween the supply container and the sample cell for a period betweenone second to thirty seconds, and measuring the pressure decrease andtemperature in the supply container and the pressure and temperature inthe sample cell as a function of time until equilibration therebydetermining a curve of sample cell pressure versus time; (c) usingnumerical analysis means to provide a computer program furthercomprising a mass balance equation of the sample cell with the diffusioncoefficient as a parameter, and using a computer with the computerprogram and an estimated value of the diffusion coefficient and themeasured pressure decrease, temperature and volume of the supplycontainer and measured volume, temperature and weight of powder in thesample cell to compute a curve of sample cell pressure versus time; (d)repeating the computation in step (c) to optimize the diffusioncoefficient parameter by finding the best fit between the computed curveof sample cell pressure versus time and the corresponding curve measuredin step (b), and using the optimized diffusion coefficient as themeasured diffusion coefficient.
 2. The method and apparatus of claim 1for measuring a molecular diffusion coefficient in a porous powder,further comprising repeating steps (b) to (d) until the pressure in thesample cell has reached a pressure beyond which no additional gas uptakeoccurs in the porous powder to get the diffusion coefficient indifferent pressure ranges, wherein the supplying of the gas to thesupply container can be omitted if the supply container pressure ishigher than the sample cell pressure.
 3. The method and apparatus ofclaim 1 for measuring a molecular diffusion coefficient in a porouspowder, further comprising using the valving means to allowcommunication between the supply container and the sample cell for aperiod less than twenty seconds.
 4. The method and apparatus of claim 1for measuring a molecular diffusion coefficient in a porous powder,further comprising providing pressure measuring means with a precisionof less than 0.3% of the reading shown by the measuring means formeasuring the pressure decrease in the supply container and the pressurein the sample cell.
 5. The method and apparatus of claim 1 for measuringa molecular diffusion coefficient in a porous powder, further comprisingproviding a pressure measuring means with a precision of less than 0.3%of the reading shown by the measuring means for measuring the pressurein the sample cell and providing a reference chamber with a constantpressure and a pressure difference measuring means with a precision ofless than 0.3% of the reading shown by the measuring means that has itshigh pressure connection in communication with the reference chamber andits low pressure connection in communication with the supply containerfor measuring the pressure decrease in the supply chamber.
 6. The methodand apparatus of claim 1 for measuring a molecular diffusion coefficientin a porous powder, further comprising providing for the diffusioncoefficient parameter in the computer program to be a function of aspace variable in the porous powder.
 7. The method and apparatus ofclaim 1 for measuring a molecular diffusion coefficient in a porouspowder, further comprising measuring an adsorption isotherm of the gasand using this in the computer program.
 8. An apparatus for measuring amolecular diffusion coefficient in a porous powder comprising: (a) asupply container of known volume; (b) a sample cell with a weighedamount of the porous powder and of known volume and isolated from thesupply container by a gas flow control means, wherein the gas flowcontrol means is in the communicate state for between one second tothirty seconds; (c) a constant temperature bath surrounding the samplecell; (d) a vacuum pump assembly in communication with the supplycontainer and sample cell through valving means and connecting conduits;(e) at least one gas supply means in communication with the supplycontainer through valving means and connecting conduits; (f) a pressuredecrease measurement means in communication with the supply container;(g) a pressure measurement means in communication with the sample cell;(h) a temperature measurement means in contact with the supplycontainer; (i) a temperature measurement means in contact with a tubingto the sample cell; (j) a computer and computer program, wherein thecomputer program comprises a mass balance equation of the sample cellwith a diffusion coefficient as an adjustable parameter formulated usingnumerical analysis means, and the computer computes equilibration curvesof the sample cell pressure versus time using the computer program, avalue of the diffusion coefficient parameter and the measured pressuredecrease, temperature and volume of the supply container and measuredvolume, temperature and weight of powder in the sample cell, and thecomputer program further comprises an optimization means for selectingthe value of the diffusion coefficient parameter that gives the best fitbetween the computed and measured curves of the sample cell pressureversus time.
 9. The apparatus of claim 8 for measuring a moleculardiffusion coefficient in a porous powder, wherein the gas flow controlmeans is in the communicate state for less than twenty seconds.
 10. Theapparatus of claim 8 for measuring a molecular diffusion coefficient ina porous powder, wherein the supply container has a volume between 100cc and 500 cc and the sample cell has a volume between 1 cc and 100 cc.11. The apparatus of claim 8 for measuring a molecular diffusioncoefficient in a porous powder, wherein the pressure measurement meansin communication with the sample cell and the pressure decreasemeasurement means in communication with the supply container have aprecision less than 0.3% of the reading shown by the measuring means.12. The apparatus of claim 8 for measuring a molecular diffusioncoefficient in a porous powder, wherein the pressure measurement meansin communication with the sample cell has a precision less than 0.3% ofthe reading shown by the measuring means and the pressure decreasemeasurement means in communication with the supply container comprises areference chamber with a constant pressure and a pressure differencemeasurement means with a precision less than 0.3% of the reading shownby the measuring means with its high pressure part in communication withthe reference chamber and low pressure part in communication with thesupply container.
 13. The apparatus of claim 8 for measuring a moleculardiffusion coefficient in a porous powder, wherein the computer programcomprises a diffusion coefficient parameter that is a function of aspace variable in the powder particle.
 14. The apparatus of claim 8 formeasuring a molecular diffusion coefficient in a porous powder, whereinthe supply container has a gas amount sufficient to increase thepressure in the sample cell to a pressure beyond which no additional gasuptake occurs in the porous powder.
 15. The apparatus of claim 8 formeasuring a molecular diffusion coefficient in a porous powder, whereinthe computer program further comprises a concurrently measuredadsorption isotherm that is used in the mass balance equation of thesample cell.